First of all it seems to treat $P_1,P_n$ different from other points. Secondly he says the euler measure of an $n$ sided region is $1-n/4$ which looks different from what Lipshitz says (which takes the accute and obtuse corners into account). Last but not least I don't understand the definition of the last term. It is defined by moving the two sides in 4 different directions in such a way that no endpoint of one is on another and then taking intersection points. It's not clear for me what these 4 directions are and why the result is not always zero.

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(1) Yes, $p_1$ and $p_n$ are treated different from other points. If you cyclically permute the indices $\{1,\ldots,n\}$ you do of course get the same answer, but it's not immediately obvious from Sarkar's formula that this is the case.

(2) The Euler measure of a region $R$ equals:$$e(R):=\chi(R)-\frac 14\#\{\text{acute corners}\}+\frac 14\#\{\text{obtuse corners}\}$$Usually one just talks about regions which are disks, so $\chi(R)=1$. Now every connected component of $\Sigma_g\setminus(\eta_1^1\cup\cdots\cup\eta_g^1\cup\cdots\cup\eta_1^k\cup\cdots\eta_g^k)$ has zero obtuse corners. With these two facts in hand one reduces to Sarkar's formula $1-\frac n4$ for an $n$-sided polygon.

(3) The arcs in $\partial_iD$ all lie inside the $\boldsymbol\eta^i$ circles, and they are considered moved in "both directions" along the $\boldsymbol\eta^i$ circles. If you indulge me as I create ugly ascii art, I can give a few examples. Here $|$ denotes $\partial_iD$ and $=$ denotes $\partial_jD$.